I'm afraid I feel old fashioned or something in this regard, but when I
have a question on translation of an word or idiomatic phrase I reach
for BAGD and LSJ. Is this cheating or something? Perhaps it does take
some of the fun out of the discussion if we regard them as the last
word. I certainly don't recommend that, although perhaps they should
have the first word?

Since no one seems to have quoted these sources, let me simply issue
their opinions into the mix of discussion: "ARCHN LAMBANEIN *begin*
[they cite Polybius, Aelian; Diogenes Laertius, Sextus Empiricus, and
Philo] LALEISQAI *be proclaimed at first* Hb 2:3, cf. IEph 19:3" (BAGD,
p. 112; the *'s mark italics [art. ARCH]). The reference in Ignatius
reads: ARCHN DE ELAMBANEN TO PARA QEWi APHRTISMENON which Lightfoot
renders "and that which had been perfected in the counsels of God began
to take effect." LSJ does not discuss this pair directly under ARCH, but
give an interesting reference to acc. ARCHN used absolutely to mean "to
begin with" or "not at all" w/ a negative citing ARCHN MHDE LABWN in
Herodotus (3.39) [which I don't have at home].

If I were inclined to pursue this, I would ask whether: (1) ARCHN
LAMBANEIN were a Hellenistic idiom coming alongside ARCOMAI + inf. as
virtually synonymous and hence of little further interest; or, (2) the
phrase has a slightly different nuance than ARCOMAI + inf. which focuses
on the inception of an action; i.e., "which was *at the very beginning*
spoken through. . . " in Heb. 2:3. I'm not defending either idea, and
the second is just a guess, but if someone wanted to do some lexical
research in TLG, this might be a fruitful inquiry.